Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Mittag-Leffler series for generating continuum sum?

I saw this:

Apparently, it seems there is a formula that can extend a Taylor series to a whole cut complex plane, called a Mittag-Leffler star of the function. This is interesting: because then perhaps maybe we could apply Faulhaber's formula to this to yield a continuum sum that works even for functions which are not entire: it appears the reason the original Faulhaber's formula (Faulhaber-on-Taylor formula) was not working is because continuum sum is a "global" operation (unlike derivative), as I mentioned in a recent post to the thread "Continuum sum formula rescued?", that is, the operation not only depends on the behavior of the function being put into it near the sum indices, but also very far away, and since a Taylor series of finite convergence radius only looks locally like the function they expand, we can explain the failure of the Faulhaber formula when applied to Taylor expansions of non-entire functions. Yet if the Mittag-Leffler series, on the other hand, converges over a whole cut plane, then this problem should not arise, as the global behavior will be correct.

If this could be done, it might then enable the usage of Ansus' sum formula for tetration to extend tetration to any complex base and height.

But the problem is I can't test it, since the description on the page is incomplete: what does the notation mean? If is the -th Taylor coefficient, then what's ? What are the magic numbers ? I can't access the references, because I don't have access to a university library.

Messages In This Thread
Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 09:14 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
Question Taylor series of i[x] Xorter 12 10,265 02/20/2018, 09:55 PM
Last Post: Xorter
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,327 01/17/2017, 05:10 AM
Last Post: JmsNxn
  Taylor series of cheta Xorter 13 10,935 08/28/2016, 08:52 PM
Last Post: sheldonison
  2015 Continuum sum conjecture tommy1729 3 3,066 05/26/2015, 12:24 PM
Last Post: tommy1729
  Another way to continuum sum! JmsNxn 6 6,011 06/06/2014, 05:09 PM
Last Post: MphLee
  [integral] How to integrate a fourier series ? tommy1729 1 2,301 05/04/2014, 03:19 PM
Last Post: tommy1729
  Continuum sum = Continuum product tommy1729 1 2,416 08/22/2013, 04:01 PM
Last Post: JmsNxn
  applying continuum sum to interpolate any sequence. JmsNxn 1 2,651 08/18/2013, 08:55 PM
Last Post: tommy1729
  Powerful way to perform continuum sum JmsNxn 7 7,369 08/12/2013, 07:17 PM
Last Post: JmsNxn
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 15,649 02/04/2012, 05:02 AM
Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)